Quantum geometry of connected state manifolds: When diabolic points act as bridges between eigenstate manifolds

The connected state manifold (CSM) framework provides a unified geometric structure that resolves metric singularities at diabolic points by gluing adjacent eigenstate manifolds together, allowing for stable geodesic computation and topological characterization.

General Theory of Quantum Geometry

The quantum geometric tensor (QGT) is an important tool for studying parametric quantum systems as it encodes both the Provost–Vallee (Fubini–Study) metric and the Berry curvature. The Provost–Vallee metric quantifies the sensitivity of eigenvectors to parameter variations in the Hamiltonian, which is strongly enhanced when spectral gaps become small and typically diverges as a gap closes. Diabolic points (DPs) are isolated degeneracies where energy surfaces meet in a double-cone structure, leading to metric singularities and difficulties for geometric description. The QGT is defined by the real part of the tensor, the Provost–Vallee metric tensor g(n)µν(α), which defines infinitesimal distances on the nth state manifold Mn via (ds(n)) squared = g(n)µν dαµdαν.

Regularization through Stretched Coordinates

To tackle the metric singularity caused by zero-dimensional DPs, a coordinate transformation into a one-dimensional boundary is used, where each DP acts similarly to an Einstein–Rosen bridge (wormhole). By introducing stretched coordinates centred at each DP, the singular point is mapped into a circular boundary (the DP horizon), and in these new local coordinates, the metric is continuous across the horizon and admits analytic continuation between adjacent eigenstate manifolds. This resulting object—the connected state manifold (CSM)—is the union of eigenstate manifolds glued along their DP horizons.

Properties of the Connected State Manifold (CSM)

The CSM is defined as the union of individual eigenstate manifolds glued along their DP horizons, allowing for adiabatic driving to occur between eigenlevels via DPs. The continuity of the metric on CSM(n,n+m), including its values over the DP horizon, follows from the continuity of the underlying Hamiltonian Hˆ(α). The topology of CSM grows complex with multiple DPs, and it can be represented as a graph where vertices represent eigenstate manifolds and edges represent DPs. The genus of the CSM structure is given by genus = −m + n+Xm−1k=nIk, number of ends = m + 1, where Ik marks the number of DPs at level k.

Geodesic Shortcuts and Stability

The CSM admits geodesic shortcuts, where boundary-value geodesics are shorter than any other geodesic confined to a single eigenstate manifold. This is achieved by using two DPs: one to traverse into an adjacent manifold and the other to return to the manifold of origin. The CSM geodesic is stable, whereas the single-manifold geodesic is infinitely sensitive to its initial condition.

Berry Phase and Nodal Lines on the CSM

The Berry phase accumulated along an adiabatic evolution can be computed using nodal line theory, which requires the system to possess an effective PT symmetry. The Berry phase formula reduces to γB = kπ mod 2π, where k is the net number of DPs enclosed by Γ. This implies that the Berry phase along any closed path depends only on the parity of nodal line crossings on this connected surface.

Numerical Regularization and Applications

In Cartesian coordinates, numerical calculations are unreliable because the metric tensor diverges near DPs, but stretched coordinates resolve this by mapping each singular point to a regular boundary. This confirms that the singularity at the DP is coordinate removable, and the two sheets Mn0 and Mn1 are now connected into a single manifold via the DP horizons. The CSM framework opens a concrete route to new geodesics by permitting trajectories to traverse DP horizons, which can be shorter than single-manifold counterparts.

Topological Classification

The topological classification of the CSM is coordinate-independent and stable against continuous real perturbations Hˆ 7→ Hˆ +δHˆ that preserve the DP count between levels n and n+ m. The CSM provides new tools for any parametric quantum system with DPs, and admits development along several directions.

How it works

The paper introduces the connected state manifold (CSM), a unified geometric framework that resolves the singular behaviour of the metric tensor at DPs by promoting each DP to a circular boundary called the DP horizon. This boundary allows for a finite continuous extension of the metric across it, enabling smooth continuation between adjacent eigenstate manifolds.

How it works

The CSM framework enables new geodesics that can traverse DP horizons, allowing adiabatic paths to cross DPs and form loops that leverage the nontrivial CSM topology. This traversal is possible because the metric divergence is removable by a change of coordinates, ensuring finite geodesic speed even at DP crossings.

How it works

The Berry phase accumulated along a path reduces to a parity count of nodal line crossings on this connected surface, providing geometric insights into the Berry phase. The topological constraints on nodal lines must start and end at points where the gauge inherently fails, such as DPs or parametric infinity.

How it works

The CSM framework addresses numerical obstructions by using stretched coordinates around individual DPs to regularize metric divergences, making the resulting metric smooth and finite across the horizon. This stability is inherited by all geometric quantities like geodesics.

How it works

The paper demonstrates that a CSM geodesic passing through two DPs in succession can be shorter than a single-state manifold geodesic, showing the advantage of this new class of trajectories in the spin-1 model. This shortcut is stable with respect to initial conditions, unlike single-manifold geodesics which are not.

How it works

The CSM framework allows for a systematic approach to computing Berry phase accumulated along adiabatic evolutions that traverse DPs in effectively PT-symmetric systems. This links the nodal line structure to the topology of the CSM and provides geometrical insights into the Berry phase.

How it works

The paper establishes a topological classification for CSMs, relating their genus and number of ends to the configuration of DPs between eigenlevels. This classification is coordinate-independent and stable against continuous real perturbations Hˆ 7→ Hˆ +δHˆ that preserve the DP count between levels n and n+ m.

How it works

The paper shows how to compute nodal lines using the adjugate matrix method, which fixes the gauge before computation and returns unnormalised eigenvectors, revealing nodal lines as a zero vector magnitude indicating gauge-choice failure. This method is used to locate these lines systematically.

How it works

The paper demonstrates that for a 3-level spin-1 Hamiltonian, the angular component of the metric tensor depends only on the two-level subsystem vector f×(φ), as in Eq. (23). The radial and off-diagonal components receive contributions from coupling to the third level and do not vanish as they do for the 2-level system.

How it works

The paper uses second-order perturbation theory to correct the angular metric tensor element g˜(0)φφ, showing that first-order perturbation theory alone fails to derive correct grr, and second-order perturbation theory corrects this.

How it works

The CSM provides a new class of geodesics by allowing trajectories to cross DP horizons, which facilitates traversals between eigenstate manifolds and permits the formation of loops leveraging the full potential of the nontrivial CSM topology. These passages facilitate traversals between eigenstate manifolds and even permit the formation of loops that leverage the full potential of the nontrivial CSM topology.

How it works

The paper shows that non-diabolic degenerate points do not serve as tunnels connecting different manifolds, and the CSM would need advanced modifications to accommodate them. The analysis shows that in the Bx direction, where the energy difference is quadratic, eigenvectors smoothly continue on the same manifold.

General Theory of Quantum Geometry

The quantum geometric tensor (QGT) is an important tool for studying parametric quantum systems as it encodes both the Provost–Vallee (Fubini–Study) metric and the Berry curvature. The Provost–Vallee metric quantifies the sensitivity of eigenvectors to parameter variations in the Hamiltonian, which is strongly enhanced when spectral gaps become small and typically diverges as a gap closes. Diabolic points (DPs) are isolated degeneracies where energy surfaces meet in a double-cone structure, leading to metric singularities and difficulties for geometric description.

1. Bold header: Metric Tensor Regularization for Quantum Systems

This allows AI systems to perform stable numerical calculations near spectral degeneracies by using stretched coordinates that regularise these divergences by mapping each singular point to a regular boundary. This translates to more reliable simulation of quantum dynamics where the standard Cartesian formulation is ill-conditioned.

2. Bold header: Geodesic Shortcut Trajectories

AI systems can utilize new trajectories that can now traverse DP horizons allowing them to perform adiabatic evolution between manifolds in ways that are not possible on single manifolds. This means the system can execute maneuvers like a geodesic starting on M1, passes through it to M0, where it continues towards DP1 through which it re-enters M1, providing a shortcut.

3. Bold header: Topological Berry Phase Computation

AI systems can compute the Berry phase for paths traversing diabolic points by using the nodal line approach to determine the accumulated phase as a parity count of nodal line crossings on this connected surface. This provides a method to calculate γB = kπ mod 2π even along paths that include DPs.

Parametric Hamiltonians often exhibit point-like spectral degeneracies, called diabolic points (DPs) or conical intersections, which can lead to singularities in the Provost-Vallee metric of eigenstate manifolds. We regularise the metric by a transformation to polar coordinates and develop a formalism in which DPs act as bridges between adjacent eigenstate manifolds, glueing them into a single connected state manifold (CSM). We characterise the topology of this structure and extend the local rules for nodal lines governing the Berry phase to global ones. On the CSM, the class of geodesics can be enlarged by those passing through the DPs. They can be stable with respect to their initial direction, and can be shorter than those constrained to a single eigenstate manifold, providing new geodesic shortcuts. Further, the CSM provides a new mechanism for Berry phase computation along paths traversing the DPs.